Composing functions, inverse functions, graphing and evaluating inverse functions, computing inverses and domain restrictions
Want a deeper conceptual understanding? Try our interactive lesson! (Plus Only)
No exercises available for this concept.
Functions can be composed by passing the output of one into the other. We use the symbol ∘, and pay close attention to the order in which functions are composed:
To find an expression for f(g(x)), substitute g(x) for x in the expression for f(x).
We can find x=f−1(b) by applying the function to both sides:
So finding f−1(b) is equivalent to solving f(x)=b.
Graphically, find f−1(b) is equivalent to being given y=b, and finding the value of x for which that is true:
Powered by Desmos
The graph of a function f shows all the points (x,f(x)). Since f−1 undoes f, its graph will show all the points (f(x),x). In other words, the x and y values are swapped.
This is equivalent to reflecting the curve y=f(x) in the line y=x:
Powered by Desmos
Since f−1 undoes f, the domain of f−1 is all the possible values f could output. That is, the domain of f−1 is the range of f.
The range of f−1 is all the possible values that could have gone into f. Thus, the range of f−1 is the domain of f.
Formally, the inverse function is such that
We call x the identity function, as I(x)=x composed with any function gives the same function.
Inverse functions f−1(x) can be found algebraically by switching x and y in the expression for f(x) and attempting to isolate y.
Recall that a function f must pass the vertical line test to guarantee that each input gives at most one output.
Since f−1 is also a function, it too must pass the vertical line test. But since the graph of f−1(x) is a reflection of the graph of f(x) in the line y=x, the graph of f(x) must then pass the horizontal line test:
Powered by Desmos
A function that passes the horizontal line test is said to be one to one - each input yields exactly one distinct output. Such a function is said to be invertible, which means f−1 exists.